Theoretical Analysis Based on Particle Electro-Mechanics for Au
Pearl Chain Formation
Minglin Li1,3,4, Fei Fei1,4, Yanli Qu1,2, Zaili Dong1, Wen J. Li1,2,* and Yuechao Wang1
1
Robotics Laboratory, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China
2
Centre for Micro and Nano Systems, The Chinese University of Hong Kong, Hong Kong, China
3
College of Mechanical Engineering, Fuzhou University, Fujian, China
4
Graduate University of the Chinese Academy of Sciences, Beijing, China
Abstract — This paper analyzes the fundamental mechanisms in
driving Au Pearl Chain Formation (PCF) based on
dielectrophoresis (DEP) force. From experimental results, the
PCF process strongly depends on the voltage and the frequency
applied on electrodes, but weakly on the sizes of particles, which
appears to be contrary to theoretical expectations. To explain the
above phenomenon, we estimated the DEP force and the
Brownian motion imposed on the Au nanoparticles, and then
investigated the AC electro-osmosis force and the electro-thermal
force which may possibly affect the PCF rate. Numerical
modeling to compare the forces is presented. By matching
experimental and numerical results, we validate the scaling laws
of the DEP force and electro-mechanics in the PCF of Au nanoparticles.
Keywords — Dielectrophoresis, Gold Nano-paritcles, Pearl
Chain Formation, Nano-assemby, Nano-manipulation
I.
INTRODUCTION
The assembly of colloidal particles into functional
structures has attracted significant interest due to its potential
nanotechnology applications [1-6], such as sensors, nanostructured films, and wires for electrical circuits. During the
last two decades, electric-field-driven movement of particles
using effects such as electrophoresis [6], dielectrophoresis [5,
7-9], and electrohydrodynamics [10], has been carried out to
assemble nanoparticles in suspensions. Diverse patterns, such
as fractal, column, and pearl chain, have been observed,
respectively [5, 11] or simultaneously [10], during these
assembly processes.
Evidence reveals that different ambient conditions result in
diverse pattern formations. For instance, surface properties
significantly affect the field-induced pattern formation [12],
with fractal and chain patterns formed by coating with
conducting and insulating films on the glass particles
suspended in liquid, respectively. In assembling gold nanoparticles into wires between coplanar electrodes with
micrometer scale gap applied by alternating electric field, the
pearl chain wires and the broad and diffusion structures formed
depending on the voltage and the frequency of the applied
electric field [3, 11].
The underlying mechanism of growth formations due to
non-uniform electric fields is considered to be of the Columbic
interactions and electrohydrodynamic effects [10, 13, 14].
Experiments have exhibited that fluid flows and dielectrophoresis predominate in certain frequencies of applied AC
electric field and certain regions of plate surface, respectively
This project is funded by the Chinese National 863 Plan (Project code:
2006AA04Z320) and the NSFC (Project code: 60635040 and 60675060).
*Contact author: wen@mae.cuhk.edu.hk. Wen J. Li is a professor at The
Chinese University of Hong Kong and an affiliated professor of the Shenyang
Institute of Automation.
[15]. In this paper, we present how the frequency of applied
AC electric field influences the velocity of gold nanoparticles
in a coplanar microelectrode system, which agrees with our
experimental observations. Following the scale laws of AC
electric-kinetics, we estimated the order of magnitude of the
various forces experienced by a particle around a simplified
ideal microelectrode structure, and determined the dominating
forces that drive the PCF process.
II.
THEORY
Dielectrophoresis (DEP) is the motion of a particle arising
from the interaction of a non-uniform electric field with the
induced effective dipole moment of the particle [7, 8]. In an

electric field E ，a dielectric particle behaves as an effective
induced dipole with dipole moment p proportional to the

electric field, that is p  E . In the presence of an electric field


gradient, the force on a dipole is F   p  E . For an isolated
homogeneous dielectric sphere with loss, the complex effective
moment is [7]


p  4m f  R3 E
(1)
Here, ω is the angular frequency of the applied field, R the
particle radius, and f (ω) the Clausius-Mossotti factor defined
as [7]
f   
 *p   m*
 *p  2 m*
(2)
where  *    j /  is the complex permittivity, j   1 , and
subscript m and p representing the surrounding media and the
particle.
The time-average DEP force on an isolated homogeneous
lossy sphere is given by [7]

2
FDEP  2m R 3 Re f  Erms
(3)
where E rms , a function of position, is the root-mean-square
(RMS) magnitude of the imposed AC electric field. The real
component of the Clausius-Mossotti factor, Re[f(ω)],
determines the frequency dependence of the average DEP
force. This frequency dependence, which depends upon the
parameters of the particle and the suspending medium, defines
the DEP spectrum and the motion of the particle. When the real
part of factor is positive, particle moves toward higher electric
field regions, and this is termed positive dielectrophoresis.
When it is negative, the particle moves toward smaller electric
field regions, and this is termed negative dielectrophoresis.
Generally, gold nanoparticles are produced by the method
of liquid chemical reduction. To prevent the particles from
aggregating, some sort of stabilizing agent that sticks to the
nanoparticle surfaces is usually added. This means that the gold
nanoparticles can not be treated as homogeneous conductive
spheres because of their coating. In order to explore the
frequency-dependent characterization of gold nanoparticles in
the manipulation of DEP, we select the thin surface layer
model [7] to calculate the DEP spectrum.
III.
SINGLE SHELL MODEL
Consider a particle with a very thin outer layer shown in
Figure 1. The particle consists of a core sphere with radius R,
permittivity  C , and conductivity  C , which is covered by a
uniform layer of thickness Δ << R, permittivity  S , and ohmic
conductivity  S   C . The particle is modeled in terms of
cS   S / , in F/m 2 
transconductance g S   S / , in S/m 2  .
surface
capacitance
and
surface
The effective complex permittivity of the particle can be
expressed as [7]
 2* 
cS* R C*
cS* R   C*
(4)
where cS*  cS  jgS /  is the complex capacitance per unit
area. Assumed that the transconductance is small enough to be
ignored, the expression for the effective permittivity can be
rewritten as [7]
Thin surface layer: cm
gm

R
c  c
External medium:  m
Figure 1 Single shell model of gold colloid.
m
III
0.5
Re[f( )]
For a metallic nanoparticle in aqueous suspensions Re[f(ω)]
≈ 1, which leads to positive dielectrohoresis. As an excellent
conductor, gold nanoparticles exposed to non-uniform field
will be attracted to intense electric field regions and accumulate
together to form fractal and chain patterns [5, 10, 11]. We note
here that the washed gold nanoparticles mainly formed fractal
pattern, and less dependent on the frequency of applied fields
[5]. Whereas, the purchased gold nanoparticles that are directly
used for the assembly of wires presented more frequencydependent and formed fractal or chain patterns in different
space regions [10].
II
I
1
0
-0.5
0
2
4
6
8
10
12
14
log( ) Hz
Figure 2 DEP spectrum of single shell sphere (Core gold particle: εc = 1
ε0; R = 100 nm; σc = 2.4×10-4 S/m. Thin surface layer: cS = 0.25 F/m; gS = 0.
Deionized water: εm = 80.18 ε0; σm = 5.5×10-6 S/m). The inset shows the
experimental results.
 2*  cS R
j C  1
j  S   C   1
(5)
Here,  S  cS R /  C and  C   C /  C . Combining Equation
(5) and Equation (2), the complex Clausius-Mossotti factor
becomes [7]:
f    
 2  m S   C m'   j  m'   m   S   1
(6)
 2 2 m S   C m'   j  m'   m  2 S   2
Here,  m   m /  m and  m'  c S R /  m . To determine the
frequency dependence of the DEP force acting on the particle,
it is necessary to obtain the real part of Equation (6).
For the simulation described below, gold colloids with the
size of 100nm suspended in deionized water are considered.
Figure 2 plots Re[f(ω)] for a set of parameters. Note three
regions of the frequency spectrum where Re[f(ω)] is roughly
constant, separated by two distinct breakpoints, which is
similar to our experiments (see reference [16] for detail), as
shown in the inset of Figure 2. According to the single shell
model of the colloid suspension, the frequency spectrum is
controlled by a number of parameters, such as the complex
dielectric constant of core particle, of the surface layer, of the
suspension medium, and the colloid radius.
We discuss below how various physical parameters
influence the value of Re[f(ω)]. The simulation results are
shown in Figure 3. Note that these parameters also influence
the magnitude of the real part of Clausius-Mossotti factor more
or less, in addition to the low or high cross-over frequency.
Table 1 Summary of the parameters affecting the DEP force spectrum.
The general trends are given for three growth characteristics within the
ranges studied. The characteristics increase (↑), decrease (↓), or remain
constant (－) as each parameters is increased. Additional characteristic
(  ) refers to less trend.
Parameters
R
cS
m
c
f low 







f high 






Magnitude 






m
c
0.4
cm =10
cm =1
0.5
cm =0.1
cm =0.01
0
-0.5
0
2
4
6
8
10
12
1
 =5.5X10
-7
1
0.2
 =5.5X10
Real component of CM factor
cm =20
Real component of CM factor
Real component of CM factor
1
-6
1
0
 =5.5X10
-5
 =5.5X10
-4
1
-0.2
1
-0.4
-0.6
 =5.5X10
-3
1
-0.8
-1
0
14
frequency (log) Hz
0.4
5
10
frequency (log) Hz
 =1
1
0.6
 =50
1
 =3X10
-3
c
c
0
 =3X10
-4
 =3X10
-5
 =3X10
-6
c
c
-0.2
c
-0.4
1
0
-0.6
0
5
10
frequency (log) Hz
15
 =500
1
 =1000 
R=50nm
0.2
0
R=10nm
-0.2
R=2nm
-0.4
R=1nm
-0.6
-0.8
0
0
0
0
5
10
frequency (log) Hz
15
5
10
frequency (log) Hz
15
0.8
0.4
Real component of CM factor
-2
Real component of CM factor
Real component of CM factor
 =3X10
0
 =100
R=100nm
0.2
1
0.5
-0.5
0
15
0
 =10
2
4
6
8
10
12
0.6
c
 =500
0.4
c
0
 =100
0.2
c
 =50
0
-0.2
c
 =1
c
0
0
0
0
-0.4
-0.6
0
14
frequency (log) Hz
Figure 3 Re[f(ω)] versus frequency by varying parameters of single shell sphere to show their influences.
The effects of these parameters on the left and right cross
over frequency and the magnitude are summarized in Table 1.
The surface capacitance and solution conductivity are of major
importance to the low critical frequency. The solution
permittivity and core particle conductivity mainly have impact
on the high critical frequency. The effects of these parameters
on the magnitude vary extensively and have some peak
maxima.
Manipulating gold colloids using DEP effects to form
nanoscale device [3], microwires [5, 13], and gold colloidal
crystals [16] have been demonstrated, hence DEP can be
regarded as a remarkably simple, highly reproducible, and low
cost nano-manipulation technology. The DEP manipulation of
micrometer particles has been extensively demonstrated in the
past decade, i.e., bioparticles, dielectric particles, and metallic
particles could all be manipulated by DEP force. However, the
effects of Brownian motion will become greater as the size of
the particle is reduced. One procedure to solve this problem is
to increase the electric field by reducing the characteristic
dimensions of electrode structure. However, the electricinduced fluid flow can be a greater limiting factor than
Brownian motion. According to the scaling law in reference
[15], it is possible to predict the feasibility of DEP
manipulation giving the size of particles.
IV.
SCALING LAWS
Consider a simplified electrode structure, as shown in
Figure 4(a). The size of the gap between the coplanar
electrodes 2r is 2 micrometer. The three-dimensional electrical
field streamlines are also presented in Figure 4(b). At the
section perpendicular to the centre of the surfaces of electrodes,
Medium
+V
Electrodes
-V
2μm gap
supplyElectrolyte
Glass substrate
~
AC power supply
supplyElectrolyt
(a)
supplyElectrolyte
e
(b)
Figure 4 (a) The schematic diagram of the electrodes. (b) ThreesupplyElectrolyt
supplyElectrolyt
dimensional
electric field streamlines across
the electrodes.
e
e
the electric field lines can be simplified as semi-circular (see
the reference [15] Fig. 1 (b)).
For gold colloids suspending in deionized water, the
magnitude of the velocity of a particle in a gravitational field is
u g  0.2
R2m g

(7)
where η is the viscosity of water (1cp),  m the density of water
(1 g/cm3), and g the acceleration of gravity (9.8 m/s2).
Assuming a Clausius- Mossotti factor of 1.0 and an electric
field given by E  V r , the magnitude of the DEP velocity
of a particle, at a distance r from the centre of gap, is
u DEP  0.03
R 2  mV 2
 r3
(8)
The magnitude of the velocity of a particle arising from
Brownian motion can be given by
10
2
10
velocity (m/s)
10
10
10
10
10
10
10
10
Electroosi. (0.1MHz)
DEP (6V)
Electroos. (1kHz)
10
5
0
-2
10
0
Electroos. (1kHz)
-4
DEP (2nm)
Brownian Electroosi. (0.1MHz)
-6
velocity (m/s)
10
-8
-10
Electrotherm. ( / << 1)
Gravity
-12
10
Brownian
-5
Electrotherm. ( / << 1)
10
-10
Electrotherm. ( / >> 1)
Gravity
Electrotherm. ( / >> 1)
-14
2nm
10nm
10
100nm
-15
-16
Buoyancy
Buoyancy
-18
10
-9
10
-8
particle size (m)
10
-7
10
10
-6
-20
10
0
10
1
10
2
voltage (V)
(a)
(b)
Figure 5 The velocity of a particle suspending in DI water versus (a) particle size, and (b) applied voltage (r = 1μm).
u Bro 
1 k BT
2 3Rt
(9)
Here, kB is Boltzman’s constant, T the absolute temperature
300 K, and t the period of observation. Consider the electricinduced fluid flows such as electrothermal flow, Buoyancy
flow, and AC electro-osmosis flow, the magnitudes of the
velocities of a particle are given by
Electrothermal (  m /  m  1 ):
5  10 4
 m mV 4  m
k r  m T
(10)
Electrothermal (  m /  m  1 ):
2.5  10 4
 m mV 4  m
k r  m T
(11)
Buoyancy:
2
2
    V gr
2  10 2  m  m
 T  k
(12)
AC Electroosmosis:
0.1

 mV 2  2
 r 1   2 2
 m r
CS
; 
C S  C D 
2 m  D
(13)
+2% per degree, 1 /  m  m / T  = -0.4% per degree, and
1 /  m  m / T  = 10-4 per degree [18].
Figure 5 shows the velocity of a particle suspending in deionized water with conductivity 5.5×10-6 S m-1, plotted (a) as a
function of particle radius and (b) as a function of voltage
applied between the microelectrodes, using the same
characteristic length r = 1 μm. Figure 5(a) shows that the
velocity due to Brownian motion is greater than that due to
DEP at 6V for a particle of approximate 2 nm radius in such
microelectrodes. This was validated in our experiments [16].
Figure 5(b) suggests that higher voltages and higher
frequencies are needed to overcome the effects of Brownian
motion and AC electro-osmosis. However, an intensive electric
field is impractical for the clearly predominant electro-thermal
effect, the electrolysis, and the corrosion of electrodes. In
addition, the frequency-dependent real part of ClausiusMossotti factor may not achieve the max value of 1 due to
constraints of electrical properties of gold colloids and
suspending solutions. Therefore, it is difficult to directly
manipulate a 2 nm gold colloid using DEP. Washing the gold
colloids before applying electric field can remove the remnants
and stabilizing agents that stick to the colloid surface to make
the gold nanoparticles become conductive spheres. However,
metallic particles, exposed on non-uniform electric field, will
aggregate to form fractal pattern [13], hence the formation of
gold particle wires becomes much more difficult to control
when the particle size becomes less than 50 nm.
(14)
where k is the thermal conductivity of water, CS the
capacitance of the Stern layer, CD the capacitance of the
diffusion layer, and D the Debye length (≈10 nm [17]), with
CD   / D . For water, k = 0.6 Jm-1s-1K-1, 1 /  m  m / T  =
V.
CONCLUSION
The frequency-dependent characteristic of a gold colloid
suspending in deionized water manipulated by DEP force was
explored. The reverse-bell-shape curve of the frequency
characteristic shows that a narrow region of frequency of
applied voltage will be the optimal condition to manipulate a
gold colloid, which was demonstrated in our experiments. The
order-of-magnitude of the various forces that are experienced
by a gold colloid in a simplified ideal system was estimated.
Results show that a gold colloid with the size of about 2 nm
will be difficult to be directly manipulated by DEP.
ACKNOWLEDGMENT
Minglin Li thanks Pansy Leung and Carmen Lau for the
discussion regarding the experimental data of the gold colloid
assembly.
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